Mixed-Effects Models, also known as hierarchical linear models or multilevel models, are a powerful statistical tool used to analyze data with complex hierarchical or nested structures. These models allow researchers and analysts to account for variability at multiple levels, making them invaluable in various fields, including psychology, epidemiology, education, and more.
Mixed-Effects Models have their roots in the field of statistics and were developed to address the limitations of traditional linear models, which assume that data points are independent and identically distributed. However, many real-world datasets violate this assumption, as they often exhibit hierarchical or clustered structures.
The key foundations of Mixed-Effects Models include:
Hierarchical Data Structures: Mixed-Effects Models are designed to handle data that are organized hierarchically or in nested structures. This hierarchy can be observed in various contexts, such as students within schools, patients within hospitals, or repeated measurements within individuals.
Random Effects and Fixed Effects: Mixed-Effects Models incorporate both fixed effects and random effects. Fixed effects represent population-level relationships, while random effects capture individual variability within the hierarchy. These random effects account for the correlations and dependencies among observations within the same cluster.
Variance Decomposition: Mixed-Effects Models enable the decomposition of variance into different levels, allowing researchers to quantify how much variability exists at each level of the hierarchy.
The Core Principles of Mixed-Effects Models
To harness the power of Mixed-Effects Models effectively, one must adhere to several core principles:
Hierarchical Structure Recognition: Identify the hierarchical or nested structure present in the data, as this will determine the appropriate specification of random effects.
Model Specification: Clearly define the fixed effects and random effects in the model. Fixed effects are typically associated with predictor variables, while random effects are associated with the levels of the hierarchy.
Estimation Methods: Employ appropriate estimation methods, such as maximum likelihood estimation (MLE) or restricted maximum likelihood estimation (REML), to estimate model parameters.
Model Assumptions: Verify the underlying assumptions of the model, including the normality of residuals and the homoscedasticity of errors.
Interpretability: Ensure that the model results are interpretable within the context of the research question and the hierarchical data structure.
The Process of Implementing Mixed-Effects Models
Implementing Mixed-Effects Models involves several key steps:
1. Data Preparation
Data Organization: Structure the dataset to reflect its hierarchical nature, with appropriate identifiers for each level.
Variable Selection: Identify the predictor and response variables of interest.
2. Model Specification
Fixed Effects: Determine which predictors are treated as fixed effects, representing population-level relationships.
Random Effects: Identify the levels of the hierarchy and specify the random effects associated with each level.
3. Model Estimation
Estimation Method: Choose the appropriate estimation method, either MLE or REML, based on the research question and the distributional assumptions of the data.
4. Model Assessment
Assumption Checking: Assess the model’s assumptions by examining residual plots and residual distributions.
Model Fit: Evaluate the goodness of fit of the model using appropriate metrics, such as the likelihood ratio test or the Akaike information criterion (AIC).
5. Interpretation and Reporting
Fixed Effects Interpretation: Interpret the estimated fixed effects, including their coefficients and significance levels.
Random Effects Interpretation: Understand the contribution of random effects to the overall variability in the data.
Contextualization: Provide context for the model results within the hierarchical data structure and the research question.
Practical Applications of Mixed-Effects Models
Mixed-Effects Models find applications in diverse fields, offering solutions to a wide range of research questions:
1. Education Research: In educational studies, researchers use mixed-effects models to account for students nested within schools, examining the impact of teaching interventions or school characteristics on student outcomes.
2. Healthcare and Epidemiology: In healthcare, mixed-effects models are applied to analyze patient data within hospitals or clinics, investigating the effects of treatments or interventions while considering individual variability.
3. Psychology: Psychologists use mixed-effects models to study repeated measurements within individuals, exploring the evolution of psychological traits over time.
4. Social Sciences: In social sciences, researchers employ mixed-effects models to analyze survey data with hierarchical structures, examining how individual responses vary across different contexts or regions.
5. Agriculture and Environmental Sciences: Mixed-effects models are used in agriculture to analyze crop yields within different agricultural plots or in environmental science to model ecological data with nested sampling designs.
The Role of Mixed-Effects Models in Research
Mixed-Effects Models play several crucial roles in research:
Capturing Hierarchical Variation: They enable researchers to account for the variability at different levels of the hierarchy, ensuring that the analyses accurately reflect the data’s structure.
Modeling Complex Relationships: Mixed-Effects Models can handle complex relationships between predictors and outcomes, including interactions and nonlinear associations.
Handling Missing Data: They are effective at handling missing data within hierarchical structures, allowing researchers to make efficient use of available information.
Generalizability: By incorporating both fixed and random effects, these models provide insights that are more generalizable to populations beyond the observed sample.
Criticisms and Controversies
Despite their many advantages, Mixed-Effects Models are not without criticisms and controversies:
Complexity: The complexity of mixed-effects models may pose challenges for interpretation, particularly when models include numerous random effects or interaction terms.
Data Requirements: Adequate sample sizes and careful consideration of the hierarchical structure are essential for reliable model estimation.
Assumption Violations: Violations of model assumptions, such as non-normality of residuals, can impact the validity of model results.
Model Selection: Deciding on the appropriate model structure, including which random effects to include, can be challenging and may require expert judgment.
Conclusion
Mixed-Effects Models are a versatile and powerful tool for analyzing hierarchical or nested data structures. They provide researchers with the means to capture complex relationships, account for variability at multiple levels, and make more accurate and generalizable inferences. Whether applied in education, healthcare, psychology, or other fields, Mixed-Effects Models enable us to delve deeper into our data, uncovering hidden insights and answering complex research questions with precision and rigor.
Key Highlights
Hierarchical Data Structures: Mixed-Effects Models are tailored to handle data organized hierarchically or in nested structures, common in fields like psychology, epidemiology, and education.
Random and Fixed Effects: These models incorporate both fixed effects (population-level relationships) and random effects (individual variability within hierarchies), capturing correlations among observations within clusters.
Variance Decomposition: Mixed-Effects Models allow researchers to decompose variance into different levels, facilitating the quantification of variability within hierarchies.
Core Principles: Adherence to principles like hierarchical structure recognition, proper model specification, appropriate estimation methods, and model assumption verification ensures effective utilization of Mixed-Effects Models.
Implementation Process: Implementing these models involves steps like data preparation, model specification, estimation, assessment, interpretation, and reporting, each crucial for accurate analysis.
Applications: Mixed-Effects Models find wide-ranging applications in fields like education, healthcare, psychology, social sciences, agriculture, and environmental sciences, addressing diverse research questions.
Roles in Research: They play crucial roles in capturing hierarchical variation, modeling complex relationships, handling missing data, and providing insights generalizable to broader populations.
Criticisms and Controversies: Challenges include model complexity, data requirements, assumption violations, and model selection difficulties, which may impact interpretation and validity.
Conclusion: Despite challenges, Mixed-Effects Models stand as powerful tools for analyzing hierarchical data, enabling researchers to uncover hidden insights and answer complex research questions with precision and rigor across various disciplines.
Convergent thinking occurs when the solution to a problem can be found by applying established rules and logical reasoning. Whereas divergent thinking is an unstructured problem-solving method where participants are encouraged to develop many innovative ideas or solutions to a given problem. Where convergent thinking might work for larger, mature organizations where divergent thinking is more suited for startups and innovative companies.
The concept of cognitive biases was introduced and popularized by the work of Amos Tversky and Daniel Kahneman in 1972. Biases are seen as systematic errors and flaws that make humans deviate from the standards of rationality, thus making us inept at making good decisions under uncertainty.
Second-order thinking is a means of assessing the implications of our decisions by considering future consequences. Second-order thinking is a mental model that considers all future possibilities. It encourages individuals to think outside of the box so that they can prepare for every and eventuality. It also discourages the tendency for individuals to default to the most obvious choice.
Lateral thinking is a business strategy that involves approaching a problem from a different direction. The strategy attempts to remove traditionally formulaic and routine approaches to problem-solving by advocating creative thinking, therefore finding unconventional ways to solve a known problem. This sort of non-linear approach to problem-solving, can at times, create a big impact.
Bounded rationality is a concept attributed to Herbert Simon, an economist and political scientist interested in decision-making and how we make decisions in the real world. In fact, he believed that rather than optimizing (which was the mainstream view in the past decades) humans follow what he called satisficing.
The Dunning-Kruger effect describes a cognitive bias where people with low ability in a task overestimate their ability to perform that task well. Consumers or businesses that do not possess the requisite knowledge make bad decisions. What’s more, knowledge gaps prevent the person or business from seeing their mistakes.
Occam’s Razor states that one should not increase (beyond reason) the number of entities required to explain anything. All things being equal, the simplest solution is often the best one. The principle is attributed to 14th-century English theologian William of Ockham.
The Lindy Effect is a theory about the ageing of non-perishable things, like technology or ideas. Popularized by author Nicholas Nassim Taleb, the Lindy Effect states that non-perishable things like technology age – linearly – in reverse. Therefore, the older an idea or a technology, the same will be its life expectancy.
Antifragility was first coined as a term by author, and options trader Nassim Nicholas Taleb. Antifragility is a characteristic of systems that thrive as a result of stressors, volatility, and randomness. Therefore, Antifragile is the opposite of fragile. Where a fragile thing breaks up to volatility; a robust thing resists volatility. An antifragile thing gets stronger from volatility (provided the level of stressors and randomness doesn’t pass a certain threshold).
Systems thinking is a holistic means of investigating the factors and interactions that could contribute to a potential outcome. It is about thinking non-linearly, and understanding the second-order consequences of actions and input into the system.
Vertical thinking, on the other hand, is a problem-solving approach that favors a selective, analytical, structured, and sequential mindset. The focus of vertical thinking is to arrive at a reasoned, defined solution.
Maslow’s Hammer, otherwise known as the law of the instrument or the Einstellung effect, is a cognitive bias causing an over-reliance on a familiar tool. This can be expressed as the tendency to overuse a known tool (perhaps a hammer) to solve issues that might require a different tool. This problem is persistent in the business world where perhaps known tools or frameworks might be used in the wrong context (like business plans used as planning tools instead of only investors’ pitches).
The Peter Principle was first described by Canadian sociologist Lawrence J. Peter in his 1969 book The Peter Principle. The Peter Principle states that people are continually promoted within an organization until they reach their level of incompetence.
The straw man fallacy describes an argument that misrepresents an opponent’s stance to make rebuttal more convenient. The straw man fallacy is a type of informal logical fallacy, defined as a flaw in the structure of an argument that renders it invalid.
The Streisand Effect is a paradoxical phenomenon where the act of suppressing information to reduce visibility causes it to become more visible. In 2003, Streisand attempted to suppress aerial photographs of her Californian home by suing photographer Kenneth Adelman for an invasion of privacy. Adelman, who Streisand assumed was paparazzi, was instead taking photographs to document and study coastal erosion. In her quest for more privacy, Streisand’s efforts had the opposite effect.
As highlighted by German psychologist Gerd Gigerenzer in the paper “Heuristic Decision Making,” the term heuristic is of Greek origin, meaning “serving to find out or discover.” More precisely, a heuristic is a fast and accurate way to make decisions in the real world, which is driven by uncertainty.
The recognition heuristic is a psychological model of judgment and decision making. It is part of a suite of simple and economical heuristics proposed by psychologists Daniel Goldstein and Gerd Gigerenzer. The recognition heuristic argues that inferences are made about an object based on whether it is recognized or not.
The representativeness heuristic was first described by psychologists Daniel Kahneman and Amos Tversky. The representativeness heuristic judges the probability of an event according to the degree to which that event resembles a broader class. When queried, most will choose the first option because the description of John matches the stereotype we may hold for an archaeologist.
The take-the-best heuristic is a decision-making shortcut that helps an individual choose between several alternatives. The take-the-best (TTB) heuristic decides between two or more alternatives based on a single good attribute, otherwise known as a cue. In the process, less desirable attributes are ignored.
The bundling bias is a cognitive bias in e-commerce where a consumer tends not to use all of the products bought as a group, or bundle. Bundling occurs when individual products or services are sold together as a bundle. Common examples are tickets and experiences. The bundling bias dictates that consumers are less likely to use each item in the bundle. This means that the value of the bundle and indeed the value of each item in the bundle is decreased.
The Barnum Effect is a cognitive bias where individuals believe that generic information – which applies to most people – is specifically tailored for themselves.
First-principles thinking – sometimes called reasoning from first principles – is used to reverse-engineer complex problems and encourage creativity. It involves breaking down problems into basic elements and reassembling them from the ground up. Elon Musk is among the strongest proponents of this way of thinking.
The ladder of inference is a conscious or subconscious thinking process where an individual moves from a fact to a decision or action. The ladder of inference was created by academic Chris Argyris to illustrate how people form and then use mental models to make decisions.
Goodhart’s Law is named after British monetary policy theorist and economist Charles Goodhart. Speaking at a conference in Sydney in 1975, Goodhart said that “any observed statistical regularity will tend to collapse once pressure is placed upon it for control purposes.” Goodhart’s Law states that when a measure becomes a target, it ceases to be a good measure.
The Six Thinking Hats model was created by psychologist Edward de Bono in 1986, who noted that personality type was a key driver of how people approached problem-solving. For example, optimists view situations differently from pessimists. Analytical individuals may generate ideas that a more emotional person would not, and vice versa.
The Mandela effect is a phenomenon where a large group of people remembers an event differently from how it occurred. The Mandela effect was first described in relation to Fiona Broome, who believed that former South African President Nelson Mandela died in prison during the 1980s. While Mandela was released from prison in 1990 and died 23 years later, Broome remembered news coverage of his death in prison and even a speech from his widow. Of course, neither event occurred in reality. But Broome was later to discover that she was not the only one with the same recollection of events.
The bandwagon effect tells us that the more a belief or idea has been adopted by more people within a group, the more the individual adoption of that idea might increase within the same group. This is the psychological effect that leads to herd mentality. What in marketing can be associated with social proof.
Moore’s law states that the number of transistors on a microchip doubles approximately every two years. This observation was made by Intel co-founder Gordon Moore in 1965 and it become a guiding principle for the semiconductor industry and has had far-reaching implications for technology as a whole.
Disruptive innovation as a term was first described by Clayton M. Christensen, an American academic and business consultant whom The Economist called “the most influential management thinker of his time.” Disruptive innovation describes the process by which a product or service takes hold at the bottom of a market and eventually displaces established competitors, products, firms, or alliances.
Value migration was first described by author Adrian Slywotzky in his 1996 book Value Migration – How to Think Several Moves Ahead of the Competition. Value migration is the transferal of value-creating forces from outdated business models to something better able to satisfy consumer demands.
The bye-now effect describes the tendency for consumers to think of the word “buy” when they read the word “bye”. In a study that tracked diners at a name-your-own-price restaurant, each diner was asked to read one of two phrases before ordering their meal. The first phrase, “so long”, resulted in diners paying an average of $32 per meal. But when diners recited the phrase “bye bye” before ordering, the average price per meal rose to $45.
Groupthink occurs when well-intentioned individuals make non-optimal or irrational decisions based on a belief that dissent is impossible or on a motivation to conform. Groupthink occurs when members of a group reach a consensus without critical reasoning or evaluation of the alternatives and their consequences.
A stereotype is a fixed and over-generalized belief about a particular group or class of people. These beliefs are based on the false assumption that certain characteristics are common to every individual residing in that group. Many stereotypes have a long and sometimes controversial history and are a direct consequence of various political, social, or economic events. Stereotyping is the process of making assumptions about a person or group of people based on various attributes, including gender, race, religion, or physical traits.
Murphy’s Law states that if anything can go wrong, it will go wrong. Murphy’s Law was named after aerospace engineer Edward A. Murphy. During his time working at Edwards Air Force Base in 1949, Murphy cursed a technician who had improperly wired an electrical component and said, “If there is any way to do it wrong, he’ll find it.”
The law of unintended consequences was first mentioned by British philosopher John Locke when writing to parliament about the unintended effects of interest rate rises. However, it was popularized in 1936 by American sociologist Robert K. Merton who looked at unexpected, unanticipated, and unintended consequences and their impact on society.
Fundamental attribution error is a bias people display when judging the behavior of others. The tendency is to over-emphasize personal characteristics and under-emphasize environmental and situational factors.
Outcome bias describes a tendency to evaluate a decision based on its outcome and not on the process by which the decision was reached. In other words, the quality of a decision is only determined once the outcome is known. Outcome bias occurs when a decision is based on the outcome of previous events without regard for how those events developed.
Hindsight bias is the tendency for people to perceive past events as more predictable than they actually were. The result of a presidential election, for example, seems more obvious when the winner is announced. The same can also be said for the avid sports fan who predicted the correct outcome of a match regardless of whether their team won or lost. Hindsight bias, therefore, is the tendency for an individual to convince themselves that they accurately predicted an event before it happened.
Gennaro is the creator of FourWeekMBA, which reached about four million business people, comprising C-level executives, investors, analysts, product managers, and aspiring digital entrepreneurs in 2022 alone | He is also Director of Sales for a high-tech scaleup in the AI Industry | In 2012, Gennaro earned an International MBA with emphasis on Corporate Finance and Business Strategy.
